1. **State the problem:** We need to expand and simplify the expression $ (3x - 2\sqrt{7})(x + \sqrt{7}) $. 2. **Recall the distributive property (FOIL method):** $ (a + b)(c + d) = ac + ad + bc + bd $. 3. **Apply FOIL to the given expression:** $$ (3x)(x) + (3x)(\sqrt{7}) + (-2\sqrt{7})(x) + (-2\sqrt{7})(\sqrt{7}) $$ 4. **Calculate each term:** - $3x \cdot x = 3x^2$ - $3x \cdot \sqrt{7} = 3x\sqrt{7}$ - $-2\sqrt{7} \cdot x = -2x\sqrt{7}$ - $-2\sqrt{7} \cdot \sqrt{7} = -2 \times 7 = -14$ 5. **Combine like terms:** $$ 3x^2 + (3x\sqrt{7} - 2x\sqrt{7}) - 14 = 3x^2 + x\sqrt{7} - 14 $$ 6. **Final answer:** $$ \boxed{3x^2 + x\sqrt{7} - 14} $$